It's 7am in Berlin and Geoffrey Gurrumul Yunupingu is standing in the green room at German TV station ZDF, an overcoat over his black leather jacket and his phone clamped to his ear. Faintly, through the earpiece, come the sounds of didgeridu, clapsticks and high, mournful singing: a week-long funeral ceremony for one of his aunts is under way on Elcho Island, off the coast of Australia's Northern Territory, and Gurrumul is checking in.

The blind singer's head cranes forward as he listens, smiling. "Wapu [brother]," he says, beckoning to his friend, helper, spokesman and double-bass player, Michael Hohnen, who is busy adjusting a nearby radiator to 32C (90F), the average temperature on Elcho (population 2,000). The two listen to the Yolngu people of north-east Arnhem Land, one of the oldest civilisations on earth.

Gurrumul, 40, is in Berlin to perform a song from his new album, Rrakala, on a popular breakfast show. A studio audience – mostly white and middle-aged except for Gurrumul's 23-year-old girlfriend, Bronwyn Gurruwiwi – are sitting around a lurid, cafe-style set. As Gurrumul waits on a couch, yawning and tapping a booted foot, Hohnen reassures the show's producer that the overcoat will come off.

That Gurrumul is back touring again after a long illness, a chronic kidney-related "lifestyle disease" that sapped his energy and forced the cancellation of 2010's US tour, is nearly as remarkable as the success story that began with his self-titled double-platinum 2009 debut. Everyone wanted a piece of him; he never complained of feeling unwell. "He thought it was jetlag," says Hohnen. "He just pushed through it."

In hospital, his condition was declared "precarious". No one knew quite what was wrong. His health, it turned out, needed managing. He overhauled his diet, gave up drinking and smoking overnight. When he started performing again in Australia last month, his Elcho-based nurse was part of his entourage, as she is now.

Few indigenous Australians have made their mark on mainstream Australia, let alone on Europe and the US. That a sightless man from a remote, disadvantaged Aboriginal community has done so on a right-handed acoustic guitar played left-handed and thus strung upside down – the way he taught himself aged six – singing in the language of his Gumatj clan, a subset of the Yolngu people, and refusing to talk to the press except through his honorary Yolngu brother Hohnen ("He's not shy; he just thinks his music speaks for itself") beggars belief. Until you hear his voice.

As ethereal and otherworldly as it is sincere and authoritative, Gurrumul has a voice that reduces people to tears (there's the story of a truck driver who heard him on Radio 2 and pulled into the hard shoulder, blubbing) and goes deep into the soul. His live shows have only heightened his mystique: between songs he sits almost completely still, his white eyes at once profound and confronting. Combine this with Australia's historically appalling treatment of its indigenous population and it's no wonder, really, that Gurrumul has come to represent everything from a soothsayer, to a conduit for old traditions, to a saint-like figure of redemption.

Spend the day with him, though, and Gurrumul is all and none of these things. His quiet self-containment makes it easy to project things on to him. Ask him a direct question, especially one with a why?, and like most Yolngu, he simply ignores you. He's a great mimic – he does a hilarious Bob Dylan – and he loves a practical joke, but he seems to spend most of his time thinking about, or speaking to people in, Elcho.

"He doesn't like being away from Elcho, but he knows it is part of his role to bring his songs to people, which means he has a livelihood," says Hohnen. "Yesterday he asked me how many shows we're doing on this tour, which is the only thing he's asked me in days. All he was interested in was getting a SIM card so he could phone home." The singer shares a house with his parents, five siblings and their children. His daughter, Jasmine, 13, lives with her mother nearby. He still padlocks his room when he's away: "Someone's got a key. Last time we forgot that the fuse box was in there and the place was in darkness for a week."

Family and community are everything to the Yolngu. Gurrumul's new single Gopuru tells of a totemic fish chasing the northwest wind, of clouds forming and ocean currents swirling: metaphors for the complex kinship patterns that exist within a system where everything is shared, decisions are taken collectively and balanda (non-Aboriginals) who spend time with the Yolngu are incorporated into the family.

As a child, Gurrumul learned to play music on a toy keyboard, then on an accordion; an aunt sang him western lullabies and traditional songs. He listened to the radio: Elvis, the Eagles, Dire Straits. "He knows all the Yolngu songs and stories; he has a lot of stuff in his head that he doesn't let out," says Hohnen.

Hohnen, a classically trained musician, met Gurrumul in the late 90s after relocating from Melbourne to Darwin, about two hours from Elcho, "to find out about the Australia ignored by colonial histories". He was stunned by Gurrumul's wide-ranging musical abilities. "He's got a classical sensibility in the way he thinks, composes and performs. There was nothing he couldn't play. I just saw this long career ahead of him opening up slowly, like a flower. I still do."

The two collaborated on Gurrumul's 2009 debut album, which was laden with awards and went on to be the No 1 world music album across Europe. There were sold-out gigs in London, Cologne, Paris; he performed with Sting on French television. I Was Born Blind featured on the soundtrack of Skins.

A second album, Rrakala, was released earlier this month, supported by the current European tour. Back in Berlin, the studio audience draws breath when Gurrumul enters, led by Hohnen. A female presenter in jeans and trainers clutches her ear-monitor, delivers some background ("So he is from birth deaf. Sorry, blind"), and asks about the song. "This is about a tuna fish riding the waves and feeling the north wind blowing on his face. Gurrumul is the tuna fish," Hohnen tells her. She looks confused.

The voice works its magic, of course. Rrakala becomes the day's second fastest rising title (beaten by the soundtrack to Dirty Dancing, which had screened the previous evening). Gurrumul's phone rings. "Lukey Boy!" he says before falling into his Gumatj language, all rolled rrs and bursts of staccato syllables ("Rrakala" is the name the Gumatj people call themselves). "Yeah, Germany," he says.

Later, we sit down to eat. A woman who has been staring at Gurrumul open-mouthed comes over and asks for his autograph ("I just saw you on the TV," she marvels. "My God, your voice!"). Bronwyn takes the woman's pen. "Bronwyn for Gurrumul," she writes.

We potter around a giant film and music store. Gurrumul and Bronwyn pick out some movies (he likes films that make Bronwyn jump), a kung-fu DVD and several CDs, replacing those that regularly go missing on Elcho. Cliff Richard and the Shadows. Abba's Gold. Iron Maiden Live in Rio. Rrakala is on the listening station, which pleases everyone no end. Shoppers double-take on seeing Gurrumul next to his own album display with the store headphones on, listening to himself playing guitar, piano and drums and singing about his ancestors, his beloved country and his totem, the saltwater crocodile. His lips move as he smiles and taps his boot, momentarily transported.

Rrakala is out now on Skinnyfish. Gurrumul plays the Barbican, London (020-7638 8891; barbican.org.uk) on 6 October.